No Solution for 3x+2y=7 in Real Numbers

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The discussion centers on the logical interpretation of the statement regarding the equation 3x + 2y = 7. The original statement suggests that there exists an x in the real numbers such that for all y in the real numbers, the equation holds true. The proposed negation is that for every x in the real numbers, there exists a y such that the equation does not hold. Participants confirm that the interpretation and negation are correct, affirming the logical structure of the statements. The conversation highlights the complexities of reading mathematical notation and the importance of precise language in logic.
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Statement:

(\exists_{x} \in R) (\forall_{y} \in R) (3x + 2y = 7)

Trying to find negation statement. This is what I think it is:

(\forall_{x} \in R)(\exists_{y} \in R) (3x + 2y ≠ 7)


Is this close?
 
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I'm having trouble reading this notation, but I'm assuming the first statement says "There exists an x in R such that for all y in R, 3x + 2y = 7."

To make this backwards, we would need to say, "For each x in R, there exists a y in R such that 3x + 2y does not equal 7," so if I have interpreted your notation correctly, I think your answer is correct.
 
Thanks, phreak. Yes, that is how it should read.
 

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